Page - 36 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Proof. Equation (7) follows from log ( 1 ρb ) = log ( P(b) ) + 〈J,b〉, andD(logP(b))=−EJ(b). The remainingof theproof is thesameas thatofProposition9. Remark15. 1. The second part of Equation (7), S(b) = log ( P(b) )−〈D(logP(b)),b〉, expresses the fact that the functions log ( P(b) ) and−S(b)areLegendre transformsof eachother: theyare linkedby the samerelation as the relationwhich linksa smoothLagrangianLandtheassociated energyEL. 2. TheLiouvillemeasureλω remains invariantunder theHamiltonianactionΦ, since the symplectic formω itself remains invariantunder that action.However,wehavenota full analogueofProposition10because themomentummap J doesnot remain invariantunder theactionΦ.Weonlyhave thepartial anologue statedbelow. 3. Legendre transformswere used byMassieu in thermodynamics in his very earlyworks [55,56], more systematically presented in [57], in which he introduced his characteristic functions (today called thermodynamicpotentials) allowingthedeterminationofall the thermodynamic functionsof aphysical systembypartial derivationsof a suitably chosencharacteristic function. Foramodernpresentationof that subject the reader is referred to [58,59],Chapter5,pp.131–152. Proposition 13. Let b ∈ G be such that the integrals defining P(b) and EJ(b) inDefinition 19 converge. ThegeneralizedGibbs state associated tob remains invariantunder the restrictionof theHamiltonianactionΦ to theone-parameter subgroupofGgeneratedbyb, { exp(τb) ∣∣τ∈R}. Proof. Theorbitsof theactiononMof thesubgroup { exp(τb) ∣∣τ∈R}ofGare the integralcurves of theHamiltonianvectorfieldwhoseHamiltonian is 〈J,b〉,whichof course is constantoneachof thesecurves. Therefore theproofofProposition10 isvalid for that subgroup. 7.2.GeneralizedThermodynamicFunctions AssumptionsMadein thisSection Notationsandconventionsbeingthesameas inSection7.1, letΩbethe largestopensubsetof the LiealgebraGofG containingallb∈G satisfyingthe followingproperties: • the functionsdefinedonM,withvalues, respectively, inRandinthedualG∗ofG, z → exp ( −〈J(z),b〉) and z → J(z)exp(−〈J(z),b〉) are integrableonMwithrespect to theLiouvillemeasureλω; • moreover their integralsaredifferentiablewithrespect tob, theirdifferentialsarecontinuousand canbecalculatedbydifferentiationunder thesign ∫ M. It isassumedinthissectionthat theconsideredHamiltonianactionΦof theLiegroupGonthe symplecticmanifold (M,ω) and itsmomentummap J are such that theopensubsetΩofG is not empty. Thiscondition isnotalwayssatisfiedwhen (M,ω) isacotangentbundle,butof course it is satisfiedwhenit isacompactmanifold. Proposition14. LetΦ : G×M→MbeaHamiltonianaction of aLie groupGona symplecticmanifold (M,ω)satisfyingtheassumptionsindicatedinSection7.2. ThepartitionfunctionPassociatedtothemomentum map J and themean value EJ of J for generalizedGibbs statesDefinition 19 are defined and continuously differentiable on the open subsetΩ ofG. For each b∈Ω, the differentials at b of the functions P and logP (whichare linearmapsdefinedonG,withvalues inR, inotherwords elementsofG∗) aregivenby DP(b)=−P(b)EJ(b) , D(logP)(b)=−EJ(b) . 36